1. Field of the Invention
The present invention relates to a method of feedback compensation for amplifiers wherein both capacitive and inductive components are used in conjunction with resistive components. It is directed to the degenerative feedback loop of an amplifier system where lost phase must be restored to the feedback signal in order to avoid oscillation.
2. Information Disclosure Statement
Before proceeding, it is important that the reader be aware of the context in which the terms "closed-loop gain," "feedback node," and "network divider ratio" will be used. Unless otherwise noted, "closed-loop gain" refers to non-inverting closed-loop gain for the purpose of this patent. Often called "noise gain" in other publications, non-inverting closed-loop gain is the pertinent term to both the stability and the gain error of any feedback amplifier system, even where said amplifier system is configured as an inverter. In publications where inverting amplifiers in particular are discussed, the term "summing node" is often substituted for the term "feedback node" because an inverting amplifier system sums the input signal with the feedback signal at its inverting input. To avoid the need for parallel terminology in this patent, the term "feedback node" is used to signify the inverting input, regardless of whether the system being discussed functions as a non-inverting amplifier or an inverting amplifier. Finally, be advised that the term "network divider ratio" has been created for this patent to denote what is referred to in other publications as "one-over-beta." The term "beta" by itself represents the quotient of the value of the divider resistor over the sum of the values of the feedback resistor and the divider resistor; its use requires repeated reference to its reciprocal, "one-over-beta." Convoluted terms tend to be counter-intuitive, e.g. substituting the phrase "not unlike" for the word "like." The term "network divider ratio," on the other hand, refers to the quotient of the sum of the values of the feedback resistor and the divider resistor over the value of the divider resistor; it need not be inverted. Incidentally, although a ratio is generally called out as two numbers, e.g. "10-to-1" or "10:1," the network divider ratio is expressed as a single number. So, if an amplifier system has a feedback resistance of 900 ohms and a divider resistance of 100 ohms, the network divider ratio is said to be "10."
The table that follows shows definitions for algebraic terms which were created to explain the background art as well as the present invention.
TABLE 1 ______________________________________ TERM DEFINITION ______________________________________ Ac1 CLOSED-LOOP GAIN Ao1 OPEN-LOOP GAIN Cfb FEEDBACK CAPACITANCE D NETWORK DIVIDER RATIO D.phi. NETWORK DIVIDER RATIO AT THE INTERCEPT .div.Err ERROR RATIO F.phi. INTERCEPT FREQUENCY Ldiv DIVIDER INDUCTANCE Rdiv DIVIDER RESISTANCE Rese ELEMENT OF SERIES EQUIVALENT RESISTANCE Rfb FEEDBACK RESISTANCE .theta..sub.Conventional PHASE ANGLE OF CONVENTIONAL FEEDBACK COMPENSATION .theta..sub.Inet PHASE ANGLE OF NETWORK CURRENT .theta..sub..rho. PHASE ANGLE OF THE PRESENT INVENTION 0.sub.Vdiv PHASE ANGLE OF DIVIDER VOLTAGE Verr ERROR VOLTAGE Vin INPUT VOLTAGE Vout OUTPUT VOLTAGE Z.sub.Cfb IMPEDANCE OF FEEDBACK CAPACITANCE Zdiv OVERALL DIVIDER IMPEDANCE Zese ELEMENT OF SERIES EQUILAVENT IMPEDANCE Z.sub.Ldiv IMPEDANCE OF DIVIDER INDUCTANCE Znet OVERALL NETWORK IMPEDANCE ______________________________________
Since their inception, solid state amplifiers have needed degenerative feedback to correct their rather large gain errors. Degenerative feedback is a means of error minification wherein the output signal is fed back to the input stage of an amplifier for comparison to the original input signal. The difference between these signals, called the "error signal," is then minified by subsequent phase-inversion and amplification. It would be even more illustrative (and more to the point) to say that the error signal is what actually drives the output to the desired amplitude; the amplifier's input stage is "satisfied" when the error signal (when multiplied by the open-loop gain) is sufficient to support the output potential. One might say that the loop is in a state of "equilibrium" at this point. It is important to remember that that which is called the "error signal" is not entirely nonlinear. Minification of said signal is still a desirable goal inasmuch as it does contain some error. As such, system performance is enhanced when a large ratio exists between open-loop gain and closed-loop gain, thereby minifying the error signal in relation to the input signal. Moreover, system performance can be predicted by computing the ratio of the error signal to the input signal. Nevertheless, from a conceptual standpoint, it is useful to point out that if a perfect amplifier with no gain error actually existed, there would be no need for large amounts of open-loop gain. For informational purposes, a complete algebraic derivation of the error ratio (.div.Err) is shown in Derivation 1. The final equation is shown in boldface at the bottom. ##EQU1##
When the desired closed-loop gain is unity, a direct connection is made between the amplifier output and the feedback node. Closed-loop gains greater than one are set by using a pair of resistors to divide the output signal for application to the feedback node, thus causing the amplifier to drive the output potential further in its quest for equilibrium. At frequencies where the open-loop gain of the amplifier system is high in relation to the closed-loop gain, the closed-loop gain is almost equal to the network divider ratio. In fact, it is common to treat these terms as though they were absolutely equal, although the closed-loop gain is always less than the network divider ratio.
Again, for informational purposes, a complete algebraic derivation of closed-loop gain (Ac1) is shown in Derivation 2. As before, the final equation is shown in boldface at the bottom. ##EQU2##
As mentioned, greater ratios of open-loop gain to closed-loop gain directly result in higher performance because the larger amounts of inverted magnification exerted on the error signal allow the amplifier to "see" and minify smaller errors. Hence, the true closed-loop gain becomes progressively closer to the network divider ratio. More importantly, any nonlinearity in the transfer function of open-loop gain becomes smaller in proportion to the input signal.
Because the open-loop gain of an amplifier decreases with increasing frequency, it will equal the network divider ratio at some point. The frequency at which this equality of open-loop gain to network divider ratio is reached is called the intercept frequency. The amplifier must not exhibit too much time delay while processing the feedback signal, otherwise its output will overshoot the desired voltage during its attempt to "satisfy" its feedback node. If the time lag of the amplifier is so large that it delays the propagation of the feedback signal by one half-cycle of the intercept frequency, the amplifier will oscillate. Despite popular belief, oscillation can only occur under said conditions.
Rather than delving into specifics prematurely, we can gain an appreciation for oscillation in the abstract with the following example. Imagine that you are colorblind and that you are staring at a flashing light. The light color, you are told, alternates between red and green; there is a brief moment during the transition of color where the light does not glow at all, hence the "flashing." There is a pushbutton connected to the light which enables you to change its color manually. You are told to press the button every time the light is about to turn green so that the light will continue to glow red. Since you are colorblind, you perform your task based on the premise that the light cycle begins with the color red and that the light color will have changed immediately following each period of darkness. In this example, the light represents the amplifier output, the pushbutton represents the feedback node, and you are the feedback loop. The color red represents an amplifier which is stable; it does not oscillate.
Now imagine that there is a time lag in the electronic processing of the pushbutton signal. In spite of your promptness in depressing the button, the light will briefly display the color green; longer electronic delays will result in longer intervals of green. The additional flashes of darkness caused by the errant appearances of the color green are no more than a minor annoyance to you. By using your natural sense of time and your familiarity with the fundamental rhythm of the flashing light, you successfully ignore the "wrong" periods of darkness and faithfully push the button on the "right" ones. In this circumstance, the "amplifier" is still stable, but not as stable as before. The brief appearances of the color green represent what is known in amplifier jargon as "overshoot." Still, the "amplifier" is not oscillating. The interval where the color red appears is three times as long as the interval where the color green appears. In other words, the color green is displayed during one-quarter of the total light cycle. Again speaking in amplifier lingo, this would compute as 25% overshoot.
Now imagine that the time delay is so long that it equals one full color interval, or one-half of the total light cycle. As before, you press the button when the light goes out. However, the electronic time delay is so great that by the time the signal reaches the light, the full green interval has run, and the light, which is about to turn red, is directed by the tardy signal to turn green again. At the end of the errant green cycle, you push the button again, sustaining a color that you thought was red. Under these conditions, the color red will never be displayed, and the light will glow green during the entire light cycle. The feedback, which was supposed to be "degenerative" to the color green, is now "regenerative," and the "amplifier" is oscillating. Notice that your colorblind eyes can only fool you into sustaining the wrong color when the pushbutton time delay is equal to one-half of the light cycle. You have no way of distinguishing a one-half cycle delay from zero delay.
Returning to the discussion of feedback amplifier systems, please note that the distinction between seconds and degrees is an important one. A time delay which is caused by a single reactance can at most only delay the phase of a signal by less than 90.degree., or one-quarter of a cycle. Lengthening the delay in seconds beyond that point only serves to attenuate the signal amplitude rather than further delaying its phase. Thus, a composite delay comprising more than one reactance is necessary for oscillation to occur in a feedback amplifier system.
To gain an appreciation of this, observe the action of a sine wave driving a capacitor through a series resistor. As the driven end of the resistor is elevated above zero volts, the capacitor, empty of charge, stifles the initial onset of the observed waveform. If the capacitor will accept a charge at all within the waveform period, it must begin before the driven end of the resistor reaches its peak. The excursion of the driven wave beyond that point towards the peak elevates the observed amplitude. The observed amplitude can certainly not be expected to rise while the amplitude at the driven end of the charging resistor is falling.
Having said this, the reader might be inclined to believe that oscillation is a rare malfunction and is of little concern. However, as desirable as it is to increase the ratio of open-loop gain to closed-loop gain, the intercept frequency also increases, making it all the more probable that the propagation delay on the part of the amplifier will be a composite one. The problem worsens if another amplifier stage is added within the same feedback loop, thus further compounding the propagation delay and increasing the risk of oscillation. Moreover, critical system parameters can vary significantly during strenuous operation, such as driving a capacitive load, making the appearance of oscillation even more likely. It is for these reasons that we can understand the inherent drawback of degenerative feedback.
Several techniques have been developed to avoid composite delay-induced oscillation. One of the more popular techniques is known as feedback compensation. The purpose of feedback compensation is to cause the feedback node of a linear amplifier to reach a desired peak voltage sooner than its output does. Recalling the previous red light/green light example, this would be analogous to being told in advance exactly what the pushbutton time delay was and then compensating for the delay by depressing the button that much earlier than usual. In an amplifier system, this is accomplished by creating an artificial peak from a portion of the output cycle which precedes the actual peak.
The conventional method for advancing the phase of a feedback signal is to place a capacitor in parallel with the feedback resistor. Typically, the capacitor has a negligible effect on the network divider ratio in the expected frequency range of input signals. However, as the intercept frequency is approached, the capacitor causes a phase-dependent reduction of the network divider ratio. Empty of charge, the capacitor can produce a voltage on the feedback node whose instantaneous value is limited only by the output amplitude itself. At some point, the capacitor's demand for current must subside to allow the feedback node voltage to decay. This is necessary to preserve the illusion that the peak was here and gone, in spite of the fact that it actually hasn't happened yet. The lower the network divider ratio, the less the artificial peak will contrast the voltage imposed by the resistive elements in the feedback network. Thus, the technique can only make a substantial contribution of phase margin to an amplifier system where the network divider ratio is sufficiently high.
By perusing the prior art, it appears that conventional feedback compensation is misunderstood, at least from a mathematical standpoint. For example, in an article in Burr-Brown.RTM. 1994 Applications Handbook, entitled "Feedback Plots Define Op Amp AC Performance", at Page 194 et seq., author Jerald Graeme asserts that the maximum capacitor value in conventional feedback compensation is limited by the potential for encountering "secondary-amplifier poles," which, stated in the more lucid terminology of this patent, refers to the excessive phase shift caused by a composite propagation delay. This statement ignores the more fundamental fact that the phase angle of current to voltage recedes as the ratio of capacitor impedance to series resistance decreases. Also, in the same article, Jerald Graeme claims that with a network divider ratio of 1001, the phase angle of conventional feedback compensation will be 45.degree.. In actuality, at said network divider ratio, a correctly optimized embodiment of conventional feedback compensation exhibits an 86.3793125913.degree. phase angle. Thus, it is appropriate to explain the correct usage of conventional feedback compensation before proceeding to a discussion of the present invention.
The phase angle of current to voltage advances when the ratio of capacitor impedance to parallel resistance decreases. Even without converting the network into equivalent series elements, it is clear that if the ratio of Z.sub.Cfb (the impedance of the feedback capacitance) to Rfb (the feedback resistance) is decreased, the phase angle will increase.
The phase angle of current to voltage recedes when the ratio of capacitor impedance to series resistance decreases. Therefore, as the ratio of Z.sub.Cfb to the combined parallel resistance of Rfb and Rdiv (the divider resistance) decreases, the phase angle decreases. At first glance, this might seem incorrect because only Rdiv is in series with Cfb. However, consider that Cfb is injecting a leading current into Rdiv as a result of either taking on or giving up charge. Two things affect the ability of Cfb to charge; one is the presence of Rdiv in series with it, supplying charge current from the output voltage. The other is the presence of Rfb, limiting the maximum possible voltage across the capacitor itself. A capacitor is regarded as "charged" when it reaches some predetermined portion of its maximum attainable voltage. Observe that the current from Rdiv originates from a voltage source which is higher than the capacitor can possibly reach: the output node. This inequality between the output voltage and the maximum attainable capacitor voltage means that the capacitor will reach a given voltage sooner than anticipated, as though Rdiv were somewhat lower in value. Indeed, the apparent value of Rdiv is scaled down from its actual value by exactly the ratio of Rfb to the sum of Rfb and Rdiv. Hence, in essence, Cfb is in series with the parallel combination of Rfb and Rdiv.
The phase angle of conventional feedback compensation (represented below by the term ".theta..sub.Conventional ") is equal to the difference between the arccotangents of the aforementioned two ratios: ##EQU3##
The optimum impedance for the compensating capacitor is equal to the feedback resistance divided by the square root of the network divider ratio: ##EQU4##
In light of the reduction in the network divider ratio, one might expect the resultant increase of the intercept frequency to degrade system stability. However, this negative effect of feedback compensation is outweighed in an amplifier system which exhibits a composite delay. Such a system exhibits a decrease in open-loop gain with increasing frequency which is steeper than the decrease in the network divider ratio.
Notwithstanding the prior art, it appears that the present invention concept wherein both capacitive and inductive components are used to advance the phase of a feedback signal is neither taught nor suggested.